Question

The integral \( \int_5^{10} \left\lfloor x \right\rfloor dx \) is equal to (where \( \left\lfloor x \right\rfloor \) denotes the greatest integer function):

• A. 55
• B. 45
• C. 35
• D. 26
• E. 5

Hint:

When solving integrals involving the greatest integer function, break the integral at each integer and evaluate the sum of the areas.

Answer 1

The Correct Option is C

Step 1: The greatest integer function \( \left\lfloor x \right\rfloor \) gives the largest integer less than or equal to \( x \). The value of \( \left\lfloor x \right\rfloor \) changes at each integer value of \( x \). Step 2: Break the integral into parts where \( \left\lfloor x \right\rfloor \) remains constant: \[ \int_5^{10} \left\lfloor x \right\rfloor dx = \int_5^6 5 dx + \int_6^7 6 dx + \int_7^8 7 dx + \int_8^9 8 dx + \int_9^{10} 9 dx. \] Step 3: Now, calculate each integral: \[ \int_5^6 5 dx = 5 \times (6 - 5) = 5, \] \[ \int_6^7 6 dx = 6 \times (7 - 6) = 6, \] \[ \int_7^8 7 dx = 7 \times (8 - 7) = 7, \] \[ \int_8^9 8 dx = 8 \times (9 - 8) = 8, \] \[ \int_9^{10} 9 dx = 9 \times (10 - 9) = 9. \] Step 4: Adding these values gives: \[ 5 + 6 + 7 + 8 + 9 = 35. \]